On Systems of Rays. 125 
sions by which these linear forms must be replaced at a principal focus or other point of 
vergency, and generally when it is proposed to determine the aberrational corrections 
of the first approximate or limiting relations, can always be obtained without difficulty 
by developing to the required order of accuracy the general and rigorous equations 
which we have given for a luminous path. An example of such deduction will occur, 
when we come to consider the theory of instruments of revolution, which on account 
of its extent and importance must be reserved for a future occasion. 
Combination of the foregoing Lap of Optics with the Undulatory Theory of Light. 
we mS By , that is, the Partial Differential Coeffi- 
cients of the First Order of the Vis tees Function V, taken with respect to 
the Final Co-ordinates, are, in the Undulatory Theory of Light, the Components 
of Normal Slowness of Propagation of a Wave. The Fundamental Formula 
(A) may easily be explained and proved by the principles of the same theory. 
The quantities o, r, v, or 
26. It remains, for the execution of the design announced at the beginning of this 
Supplement, to illustrate the mathematical view of optics proposed in this and in for- 
mer memoirs, by connecting it more closely with the undulatory theory of light. 
For this purpose we shall begin by examining the undulatory meanings of the 
symbols o, 7, v, of which, in the present Supplement, we have made so frequent a use, 
and which we have defined by the equations (2), 
her 8 oF 
~~ Sn? ay? Paps ee 
V being the undulatory time of propagation of light of some given colour, from 
some origin v, 7,2, toa point x, y, z, through any combination of media. It is 
evident that these quantities , 7, v are proportional to the direction-cosines of the 
normal to the wave for which the time V’ is constant, and which has for its differen- 
tial equation 
= oon + rey + vez ; (A*®) 
and if, as in the second number, we denote (0? + 7° + v°)~? by w, these direction-cosines 
themselves will be ow, rw, vw ; and w will be the normal velocity, because the infinite- 
simal time 8V, during which the wave propagates itself in the direction of its own 
normal through the infinitesimal line 8/, from the point , y, z, to the point 7 +ow.0/, 
Yt7w.dl, 2+vw.0l, is 
8V =0.0w.81 +7.7w.8l + v.vw.0l= 5 al: (B") 
VOL. XVII. 2K 
